The fire department has two classes of employees, responders

Question

The fire department has two classes of employees, responders and administrators. The department requires a physical for each new employee, unless that employee has had a physical in the previous year or is an administrator. Last month, the fire department required physicals of 6 new employees. If 3/4 of the new employees are responders, how many new employees does the department have?

(1) 1/6 of the new employees are administrators who have had a physical in the previous year.
(2) 1/3 of the new responders have had a physical in the previous year.

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Official Answer

B

(1) 1/6 of the new employees are administrators who have had a physical in the previous year.
(2) 1/3 of the new responders have had a physical in the previous year.

Bunuel · Answer

The fire department has two classes of employees, responders and administrators. The department requires a physical for each new employee, unless that employee has had a physical in the previous year or is an administrator. Last month, the fire department required physicals of 6 new employees. If 3/4 of the new employees are responders, how many new employees does the department have?

(1) 1/6 of the new employees are administrators who have had a physical in the previous year.
(2) 1/3 of the new responders have had a physical in the previous year.

Show Answer

Official Answer

B

Since out of new employees only responders who didn't have a physical in the previous year will need a physical, then the number of new repsonders who didn't have a physical in the previous year = 6.

(1) 1/6 of the new employees are administrators who have had a physical in the previous year. Not sufficient.

(2) 1/3 of the new responders have had a physical in the previous year. This implies that 2/3 of the new responders have NOT had a physical in the previous year. As 3/4 of the new employees are responders, then 3/4*(# of new employees)=(# of new responders) and 2/3*(# of new responders)=6 --> 2/3*3/4*(# of new employees)=6 --> (# of new employees)=12. Sufficient.

Answer: B.

boggler · Answer

Statement #1: 1/6 of the new employees are administrators who have had a physical in the previous year. 
New administrative employees do not need physicals, regardless of whether they have had a physical in the past year or not. All new employees needing new physicals are responders. This information is irrelevant. 
This statement, alone and by itself, is insufficient.

Statement #2: 1/3 of the new responders have had a physical in the previous year. 
There are six new employees that needed physicals, and all of them must be new responders. This six is the one-third of responders that needed physicals, so 2/3, 12 new responders didn't need physicals. There are 18 new responders in total, which accounts for 3/4 of all new employees, so there are 24 new employees. 
This statement, alone and by itself, is sufficient.

Hence B.

bionication · Answer

New employees can include on of the following three groups:
a) new admins - DON'T need physical
b1) new responders - received physical last years - DON'T NEED physical this year
b2) new responders - did NOT receive physical last year - NEED physical this year

From the stem, we know the following:
-25% of the new employees are admins and 75% of new employees are responders
-6 new physicals were conducted (group b2 above)

Stem B states 1/3 of new responders are in group b1 above, so 2/3 must be in group b2.
If b2 = 6, then b1 = 3 
b2+b1=9 (all responders)
If 9 responders constitute 75% of new employees, then the total number of new employees is 12.

Hence, B alone is sufficient.

Please see attached diagram for a visual of the employees breakdown.

SatvikVedala · Answer

From the question it is clear that we have (3/4)*N as responders

So, (1/4)*N are Administrators

We are given (3/4)*N - Last Year - Administrators = 6

From Statement 1, (1/4)*N = (1/6)*N + ?

From Statement 2, (1/3)*R have physical in last year
Last Year = (1/3)*R = (1/3)*(3/4*N) = (1/4)*N
Administrators = (1/4)*N

Therefore, (3/4)*N - (1/4)*N - (1/4)*N = 6
N = 6*4 = 24

Demonicgod · Answer

Given:
 
 There are  two classes  of employees:  responders  and  administrators . 
 The  department requires a physical  for each new employee  unless :
 
 that employee has had a physical in the previous year,  or  
 that employee is an administrator.  
 
 Last month, the department  required physicals for 6 new employees.  
  3⁄4 of the new employees are responders.  
 We are asked:  How many new employees does the department have in total?   
Let’s denote:
 
 Total new employees = NNN 
 Responders = 34N\frac{3}{4}N43N 
 Administrators = 14N\frac{1}{4}N41N  
 
Step 1: Who needs physicals?
Only  responders who have NOT had a physical in the previous year  require one.
So, the 6 people who required physicals = responders without recent physicals.
Thus:
Responders needing physicals=6	ext{Responders needing physicals} = 6Responders needing physicals=6
If RpR_pRp = fraction of responders who had a physical in the previous year,
 then the remaining (1−Rp)(1 - R_p)(1−Rp) fraction need new physicals.
So:
(1−Rp)(34N)=6(1 - R_p)\left(\frac{3}{4}Night) = 6(1−Rp)(43N)=6
 
We’ll use this equation with each statement to see if NNN can be found.
 
Statement (1):
 
This means: 16N\frac{1}{6}N61N are administrators  and  already had a physical.
But administrators  never need a physical  anyway (regardless of whether they had one in the previous year).
 So this information doesn’t affect how many  responders  needed a physical.
It tells us something about administrators, not responders.
 Since our equation involves only responders, this does  not  help determine RpR_pRp or NNN.
✅  Statement (1) alone is insufficient. 
 
Statement (2):
 
That means Rp=13R_p = \frac{1}{3}Rp=31.
Plug into our main equation:
(1−13)(34N)=6(1 - \frac{1}{3})\left(\frac{3}{4}Night) = 6(1−31)(43N)=6
Simplify:
23×34N=6\frac{2}{3} 	imes \frac{3}{4}N = 632×43N=6 12N=6\frac{1}{2}N = 621N=6 N=12N = 12N=12
✅  Statement (2) alone is sufficient. 
 
Statement (1) + (2):
Since (2) alone was sufficient, combining doesn’t change that.
 
✅  Final Answer: (B)  —  Statement (2) alone is sufficient, but statement (1) alone is not.

The fire department has two classes of employees, responders

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The fire department has two classes of employees, responders

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